Exponential number of stationary solutions for Nagumo equations on graphs

被引：21

作者：

Stehlik, Petr ^{[1
,2
]}

机构：

[1] Univ West Bohemia, Fac Sci Appl, Dept Math, Univ 8, Plzen 30614, Czech Republic

[2] Univ West Bohemia, Fac Sci Appl, NTIS, Univ 8, Plzen 30614, Czech Republic

来源：

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | 2017年 / 455卷 / 02期

关键词：

Reaction-diffusion equation; Graphs; Graph Laplacian; Variational methods; Bifurcations; REACTION-DIFFUSION SYSTEMS; LATTICE DYNAMICAL-SYSTEMS; DIFFERENCE-EQUATIONS; TRAVELING-WAVES; DISCRETE; EXISTENCE; NETWORKS; MODELS; CHAOS;

D O I：

10.1016/j.jmaa.2017.06.075

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We study the Nagumo reaction diffusion equation on graphs and its dependence on the underlying graph structure and reaction diffusion parameters. We provide necessary and sufficient conditions for the existence and nonexistence of spatially heterogeneous stationary solutions. Furthermore, we observe that for sufficiently strong reactions (or sufficiently weak diffusion) there are 3(n) stationary solutions out of which 2(n) are asymptotically stable. Our analysis reveals interesting relationship between the analytic properties (diffusion and reaction parameters) and various graph characteristics (degree distribution, graph diameter, eigenvalues). We illustrate our results by a detailed analysis of the Nagumo equation on a simple graph and conclude with a list of open questions. (C) 2017 Elsevier Inc. All rights reserved.

引用

页码：1749 / 1764

页数：16

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